In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.

A topological space $X$ is totally Brown if for each $n\in \mathbb{N}\setminus \left\{1\right\}$ and every nonempty open subsets $U_1,U_2,...,U_n$ of $X$ we have ${\mathrm{cl}}_X\left(U_1\right)\cap {\mathrm{cl}}_X\left(U_2\right)\cap \cdots \cap {\mathrm{cl}}_X\left(U_n\right)\ne \varnothing$. Totally Brown spaces are connected. In this paper we consider a topology ${\tau }_S$ on the set $\mathbb{N}$ of natural numbers. We then present properties of the topological space $(\mathbb{N},{\tau }_S)$, some of them involve the closure of a set with respect to this topology, while others describe subsets which are either totally Brown or totally separated. Our theorems generalize results proved by P. Szczuka in 2013, 2014, 2016 and by...

We show that the sequence of mantissas of the primorial numbers Pₙ, defined as the product of the first n prime numbers, is distributed following Benford's law. This is done by proving that the values of the first Chebyshev function at prime numbers are uniformly distributed modulo 1. We provide a convergence rate estimate. We also briefly treat some other sequences defined in the same way as Pₙ.

A topological space $X$ is totally Brown if for each $n\in \mathbb{N}\setminus \left\{1\right\}$ and every nonempty open subsets $U_1,U_2,...,U_n$ of $X$ we have ${\mathrm{cl}}_X\left(U_1\right)\cap {\mathrm{cl}}_X\left(U_2\right)\cap \cdots \cap {\mathrm{cl}}_X\left(U_n\right)\ne \varnothing$. Totally Brown spaces are connected. In this paper we consider the Golomb topology ${\tau }_G$ on the set $\mathbb{N}$ of natural numbers, as well as the Kirch topology ${\tau }_K$ on $\mathbb{N}$. Then we examine subsets of these spaces which are totally Brown. Among other results, we characterize the arithmetic progressions which are either totally Brown or totally separated in $(\mathbb{N},{\tau }_G)$. We also show that $(\mathbb{N},{\tau }_G)$ and $(\mathbb{N},{\tau }_K)$ are aposyndetic. Our results...

Consider a recurrence sequence ${\left(x_k\right)}_{k\in \mathbb{Z}}$ of integers satisfying $x_{k+n}=a_{n-1}{x}_{k+n-1}+...+a₁x_{k+1}+a₀x_k$, where $a₀,a₁,...,a_{n-1}\in \mathbb{Z}$ are fixed and a₀ ∈ -1,1. Assume that $x_k>0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k₀}<0$ then for each negative integer -D there exist infinitely many rational primes q such that $q|x_k$ for some k ∈ ℕ and (-D/q) = -1.

We are interested whether there is a nonnegative integer $u_0$ and an infinite sequence of digits $u_1,u_2,u_3,\cdots$ in base $b$ such that the numbers $u_0{b}^n+u_1{b}^{n-1}+\cdots +u_{n-1}b+u_n,$ where $n=0,1,2,\cdots ,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S.$ If any such sequence contains infinitely many elements divisible by at least one prime number $p\in S,$ then we call the set $S$ unavoidable with respect to $b$. It was proved earlier that unavoidable sets in base $b$ exist if $b\in \{2,3,4,6\},$ and that no unavoidable set exists in base $b=5.$ Now, we prove...